Economics 11. Caltech Spring Problem Set 3

Transcription

1 Economics Caltech Sping 200 Poblem Set Homewok Policy: Study You can study the homewok on you own o with a goup of fellow students. You should feel fee to consult notes, text books and so foth. The quiz will be available Wednesday at 5pm. Following the Hono code, you should find 20 minutes and do the quiz, by youself and without using any notes. Pape and pen should be all you need. Then tun it in by Thusday 5pm. (dop off in box in font of axte ). It will include one question fom each section The answes to the whole homewok will be available Fiday at 2pm. Definitions Please explain each tem in thee lines o less! Isoquant An isoquant is a contou line dawn though the set of points at which diffeent combinations of two inputs yield a given quantity of output. The typical downwad slope of the isoquants eflects the fact that adding one input while holding the othe constant eventually leads to deceasing maginal output. Isocost An isocost line epesents a combination of inputs which all cost the same amount, i.e., that hold the total expenditue on inputs constant. The slope of the isocost is the negative atio of the input pices. Maginal ate of technical substitution Amount of one input needed to make up fo a decease in anothe input and hold output constant. Moe pecisely, the maginal ate of technical substitution between inputs L and, is the ate at which units of input L can be substituted fo units of input while keeping output constant.

2 Economy of scale An economy of scale aises when an incease in output educes aveage costs. Economies of scale can be accomplished because as poduction inceases, the cost of poducing each additional unit falls. Edgewoth box The edgewoth box is a gaphical epesentation of the exchange poblem faced by the paties in a two peson, two good exchange economy. It allows a simple way of solving and visualizing thei exchange poblem. Paeto efficient allocation A Paeto efficient allocation is such that it is impossible to make one peson bette off without making the othe (o, moe geneally, someone else) wose off. It is woth noting that a Paeto efficiency allocation does not necessaily imply a fai o socially desiable distibution of esouces, since the concept makes no statement about equality o the oveall well being of a society Contact cuve Is the set of Paeto efficient allocations in an economy. Evey point on this cuve maximizes one peson s utility given the othe, and they ae chaacteized by the tangencies in the isoquants. Pice system A pice system involves a specific pice fo tading good Y fo good X, and vice vesa, that is available to both paties in a two peson, two good exchange economy. Wod poblems Please explain each question in a few sentences. State the Fist Welfae Theoem, and descibe in wods how the allocation would look like in the Edgewoth box fo a two peson, two good exchange economy. The FistWelfae Theoem states that evey competitive equilibium is Paeto Efficient. Gaphically, this implies that in an Edgewoth box, the equilibium allocation must be on the contact cuve. A fim s maginal ate of technical substitution captues the fim s willingness to tade capital fo labo, but not the degee to which capital and labo ae substitutes in supply. Tue o false? This statement is tue. The MRTS is the absolute value of the slope of the isoquant, so it measues the amount of capital that the fim is willing to give up fo one unit of labo, in ode to maintain the same level of poduction. This is not the same thing as measuing the degee to which capital and labo ae substitutes o complements.

3 Assume that, in a two peson ( and 2), two good (x and y), exchange economy, allocation A, (xa, ya, xa2, ya 2 ), and allocation, (x, y, x2, y2 ), ae both Paeto efficient. Explain why, if consume pefes allocation A, then consume 2 must pefe allocation. If consume 2 did not pefe allocation i.e., she pefeed A o would be indiffeent between A and then by moving fom to A, 2 would make consume stictly bette off without huting heself. That would mean that allocation is not Paeto efficient. Howeve, we know that is Paeto efficient. Thus, consume 2 must pefe allocation. Anothe way to see this is by dawing an Edgewoth ox plotting allocation and the tangent indiffeence cuves of consume and consume 2. ecause allocation A is pefeed by consume, it must lie above consume s indiffeence cuve. Howeve, all of those points lie on a lowe indiffeence cuve fo consume 2. A consume in an exchange economy will always be bette off at an allocation which is Paeto efficient than at an allocation which is not Paeto efficient. Tue o false? False. A Paeto efficient allocation is not necessaily an allocation that eveyone will pefe ove non Paeto efficient allocations. In a two peson, two good exchange economy, explain why both consumes maginal ates of substitution ae equal at evey point on the contact cuve. In an Edgewoth box diagam, the contact cuve is the set of points whee the indiffeence cuves of the two individuals ae tangent. We know that the maginal ate of substitution is equal to the (negative) slope of the indiffeence cuves. Also, when two cuves ae tangent at a point, thei slopes ae equal at that point. Thus, by defining the contact cuve as a set of indiffeence cuve tangencies, the maginal ates of substitution between the two goods ae equal fo the two individuals given we assume convex indiffeence cuves. Technical poblems Assume a two peson, two good exchange economy. M. lue s utility function fo goods x and y is given by: u x y A M. Red s utility function fo goods x and y is given by: u x y. lue has 20 ounces of x and 0 ounces of y. Red has 5 ounces of x and 5 ounces of v. a) Calculate the maginal ate of substitution (MRS) of y fo x fo both consumes at the endowment point. The maginal ate of substitution (MRS) of y fo x is the amount of y a consume is willing to lose to obtain an exta unit of x: u/ x MRS. u/ y A

4 Fo lue, the MRS is then: MRS x y 7 y. Since, at the endowment, x y x 70 7 x 20, y 0, we have that MRS Analogously, in the case of M. Red, the maginal ate of substitution of y fo x at the y endowment is: MRSR x b) Daw an Edgewoth box showing the endowment and indiffeence cuves of the consumes. 5 Red 0 5 lue 20 c) Assume that lue and Red can tade at a maket pice as pice takes. Assume that x is the numeaie (i.e., px ), what is the pice of y? What is the final allocation of x and y? u/ x px Fom the consume s utility maximization poblem, the condition in ou u/ y py execise yields: 0.7M M. lue ypy 0.xpx 0.7( M pxx) 0.xpx x p M. Red - R 0.9M 0.9ypy xpx 0.9( M pxx) xpx x. p Given the satisfaction of the budget constaint, the oiginal endowments, and the fact that px, the equations above become: x x

5 p y p R y x, x. Hence, the maket equilibium condition is: py py py 5, p 0.66 with y x , x Analogously, y 9.9, y 5.6. R d) Show the tading in the diagam fom pat b Red lue Assume that a fim s poduction function is given by Y f L, 6L. a) Deive the expession fo the maginal ate of technical substitution (MRTS) fo this poduction function. Recall that the maginal ate of technical substitution is the ate at which units of input L can be substituted fo units of input while keeping output constant. As such, it is the (negative) of the slope of the isoquant. We can deive the maginal ate of technical substitution using two altenative but equivalent pocedues: a.) Given the fim s poduction function, we can wite the isoquant equation as:

6 Y Y L / 6L 6. Thus, d dl Y L 6 2. Substituting 6 L fo Y in this equation, we get / / d 6 L 2 L dl 6 L, and thus d MRST dl L. a.2) Similaly, fom the definition of MRTS, it can be calculated as the atio of maginal MPL f L, poducts of the inputs and L. That is, MRTS, whee MPL, MP L f L, MP. In the case of the poduction function f L 2/ / MP 2 L. / 2/ 2 L Hence, MRTS. 2/ / 2 L L, 6 L, / 2/ MPL 2 L, Of couse, the equivalence between a. and a.2 is evident when noting that, fom Y f L, 6L, we have: f L, f, L f, L d dy d dl 0 L L dl f L, b) Let the wage ate w be $4 and the ental ate of capital be $6. Assuming that the fim is poducing 54 units of output, what will be the cost-minimizing combination of labo and capital used by the fim? What will be the fim's total cost of poducing 54 units of output? The fim is seeking to minimize it s total costs, TC wl, subject to is, the fim s poblem can be witten as: Q 6L. That mintc wl s. t. Q 6L, so that the Lagangian function is wl [ Q 6 L ]. The fist-ode conditions ae

7 2 0 w 2L 0 L L 0 0 Q 6L Solving and fo, and setting them equal yields w 2 2L 2 2 L w 2 2L 2 2 L w L wl Substituting into yields / 2 2 wl Q * Q6L L, which is the input demand fo 6 w labo. Analogously, the input demand fo capital is wl * w Q Q w 6 w 6. Hence, given Q 00, =40, w=0, we have 2 2 * 54 6 L , * , with total poduction costs: 6 6 TC wl 4(54) 6(.5) $42. Note also that / / d 6 L 2 L dl 6 L d w Note also that, fom a,, consistent with the fact that cost minimization dl L 4 equies a tangency between the isoquant and the isocost. c) Deive the expessions fo the shot-un aveage cost of poduction. Does the fim have deceasing etuns to scale? The fim s shot-un aveage cost of poduction is:

8 wl SRAC. Q Substituting in the expessions fo conditional facto demands deived in b), we obtain Q 2 ( w) 2 /2 /2 Q Q w w wl 6 w 6 6 2Q w SRAC /2 Q Q Q 6 Note, then, that SRAC incease as output inceases, implying deceasing etuns to scale.